Question

Find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. use a graphing utility to confirm your results.

Answer 1

Step 1:

The polar equation is .

Convert the equation into the conic form ,

where is the eccentricity and is the distance between the focus(pole) and the directrix.

Compare with .

Since , the equation represents an ellipse.

Step 2:

Graph the above polar equation using some polar coordinates.

Construct a table for different values of .

Graph:

Draw the polar coordinate plane.

Plot the  polar coordinates found in the table.

Connect the points with smooth curve.

Observe the graph: The ellipse eccentricity and distance .

Solution:

Eccentricity .

Distance .